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by (G 0, G) system of imprimitivity we mean a system of imprimitivity for G o based on G, acting in some separable Hilbert space ~H. In this paper we show that each (N, K) system of imprimitivity (V, E) gives rise to an (R, B) system of imprimitivity (lY, ~). If U denotes the unitary group (indexed by/~ =F/F0) associated with E, and F denotes the spectral measure of V (defined on Borel subsets of T, the circle group), then (U, F) is a (/~, T) system of imprimitivity. We show that (U, F) gives rise in a natural way to a (F, R) system of imprimitivity (~, P), and that every (F, R) system of imprimitivity is equivalent to a system of imprimitivity (~, P). Finally if ~ denotes the unitary group indexed by F with spectral measure E and _F the spectral measure of 17, then (U, F) and (U, ~) are equivalent systems of imprimitivity. We thus complete the circle of ideas involved in Gamelin's work.

On systems of imprimitivity on locally compact abelian groups with dense actions